Question

Consider the equation $(1+a+b{)}^2=3(1+a^2+b^2)$, where $a,b$ are real numbers. Then

• A. there is no solution pair $(a,b)$
• B. there are infinitely many solution pairs $(a,b)$
• C. there are exactly two solution pairs $(a,b)$
• D. there is exactly one solution pair $(a,b)$

Answer 1

The correct option is D there is exactly one solution pair $(a,b)$

$(1+a+b{)}^2=3(1+a^2+b^2)\Rightarrow 1+a^2+b^2+2ab+2a+2b=3+3a^2+3b^2\Rightarrow a^2-2ab+b^2+a^2-2a+1+b^2-2b+1=0\Rightarrow (a-b{)}^2+(a-1{)}^2+(b-1{)}^2=0\text{Since, all square terms will be non-negative.}\text{Hence, all must be zero.}\therefore a=b=1,\text{Hence, only one possible pair of solution.}$